T1 and E1 bit rate and bandwidth; DPCM encoder

Bit rate and bandwidth of T1 and E1

Both systems sample speech at $f_s = 8$ kHz with 8 bits per sample.

T1: 24 channels + 1 framing bit:

$$R_b = (24 \times 8 + 1) \times 8000 = 193 \times 8000 = 1.544\ \text{Mb/s}$$

E1: 32 slots (30 voice + 2 overhead):

$$R_b = 32 \times 8 \times 8000 = 256 \times 8000 = 2.048\ \text{Mb/s}$$

Bandwidth. The minimum (Nyquist) bandwidth for binary transmission is $B = R_b/2$. With practical NRZ/AMI pulses, the first-null bandwidth is about $R_b$.

DPCM encoder

Differential PCM uses the fact that speech samples next to each other are strongly correlated. It does not quantise each sample directly. It quantises the difference between the sample and a prediction made from earlier samples. The difference is smaller than the sample, so fewer bits are needed for the same quality.

 m[n] ->(+)--e[n]-->[Quantizer]--eq[n]--+-->[Encoder]-> bits
        ^-                              |
        |                               v
        |                   m^[n] ---> (+)
        |                     ^         | mq[n]
        +-------- m^[n] --[Predictor]<--+

• Prediction error: $e[n] = m[n] - \hat m[n]$.
• Quantised error: $e_q[n] = e[n] + q[n]$.
• Predictor input: $m_q[n] = \hat m[n] + e_q[n] = m[n] + q[n]$.

The predictor (a linear filter on past values of $m_q$) works on the quantised samples. The receiver has the same predictor, so encoder and decoder stay in step and quantisation errors do not build up.

The SNR improves by the prediction gain $G_p = \sigma_m^2 / \sigma_e^2$. DPCM saves about 1–2 bits per sample compared with PCM, e.g. 48–56 kb/s instead of 64 kb/s for speech. Adaptive DPCM (ADPCM) goes further, to 32 kb/s.